Can I put business cards in mailboxes if I also include postage?

Only if the cards actually travel through the postal service. Sticking a stamp on a card and dropping it into someone's mailbox yourself is still a violation, because a mailbox is reserved for mail that a carrier processes and delivers. To reach mailboxes legally, the pieces have to enter the mail stream properly.

• Postage alone does not make hand-placing cards in a mailbox legal. A carrier must deliver them.
• Use a saturation mailing program to blanket a neighborhood by route without needing individual addresses.
• Standard addressed mail works when you have a mailing list to target.
• Mailable formats like postcards are built for this and carry your message straight to the box.